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API

Most of the user-facing methods are extensions of functions declared in LinearAlgebra: eigen!, schur!, ordschur!, eigvals!.

Additional exported functions are as follows.

Balancing

GenericSchur.balance! — Function
balance!(A; scale=true, permute=true) => Abal, B::Balancer

Balance a square matrix so that various operations are more stable.

If permute, then row and column permutations are found such that Abal has the block form [T₁ X Y; 0 C Z; 0 0 T₂] where T₁ and T₂ are upper-triangular. If scale, then a diagonal similarity transform using powers of 2 is found such that the 1-norm of the C block (or the whole of Abal, if not permuting) is near unity. The transformations are encoded into B so that they can be inverted for eigenvectors, etc. Balancing typically improves the accuracy of eigen-analysis.

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Converting a real quasi-Schur to a true complex Schur object

GenericSchur.triangularize — Function
triangularize(S::Schur{T}) => Schur{complex{T})

convert a (standard-form) quasi-triangular real Schur factorization into a triangular complex Schur factorization.

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Obtaining eigenvectors from a decomposition

LinearAlgebra.eigvecs — Method
eigvecs(S::Schur{Complex{<:AbstractFloat}}; left=false) -> Matrix

Compute right or left eigenvectors from a Schur decomposition. Eigenvectors are returned as columns of a matrix, ordered to match S.values. The returned eigenvectors have unit Euclidean norm, and the largest elements are real.

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Eigenvalue condition numbers

GenericSchur.eigvalscond — Function
eigvalscond(S::Schur,nsub::Integer) => Real

Estimate the reciprocal of the condition number of the nsub leading eigenvalues of S. (Use ordschur to move a subspace of interest to the front of S.)

See the LAPACK User's Guide for details of interpretation.

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eigvalscond(S::GeneralizedSchur, nsub) => pl, pr

compute approx. reciprocal norms of projectors on left/right subspaces associated w/ leading nsub×nsub block of S. Use ordschur to select eigenvalues of interest.

An approximate bound on avg. absolute error of associated eigenvalues is ϵ * norm(vcat(A,B)) / pl. See LAPACK documentation for further details.

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Subspace conditioning

GenericSchur.subspacesep — Function
subspacesep(S::Schur,nsub::Integer) => Real

Estimate the reciprocal condition of the separation angle for the invariant subspace corresponding to the leading block of size nsub of a Schur decomposition. (Use ordschur to move a subspace of interest to the front of S.)

See the LAPACK User's Guide for details of interpretation.

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Circumventing normal dispatch

Mainly for testing purposes, one can invoke the generic methods implemented here for LAPACK-compatible matrix types with these methods.

GenericSchur.gschur! — Function

gschur!(A::StridedMatrix) -> F::Schur

Destructive version of gschur (q.v.).

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gschur!(H::Hessenberg, Z) -> F::Schur

Compute the Schur decomposition of a Hessenberg matrix. Subdiagonals of H must be real. If Z is provided, it is updated with the unitary transformations of the decomposition.

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GenericSchur.gschur — Function

gschur(A::StridedMatrix) -> F::Schur

Computes the Schur factorization of matrix A using a generic implementation. See LinearAlgebra.schur for usage.

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GenericSchur.geigen! — Function

geigen!(A, alg=QRIteration(); sortby=eigsortby) -> E::Eigen

Computes the eigen-decomposition of a Hermitian (or real symmetric) matrix A using a generic implementation. Currently alg may be QRIteration() or DivideAndConquer(). Otherwise, similar to LinearAlgebra.eigen!.

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Locally defined exceptions

Note: these are currently peculiar to this package but replacement by upstream analogues will not be considered breaking.

GenericSchur.IllConditionException — Type
IllConditionException

Exception thrown when argument matrix or matrices are too ill-conditioned for the requested operation. The index field may indicate the block where near-singularity was detected.

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